JP2010287135A - Data classification device, data classification method and program making computer execute the method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To perform high-speed multi-class classification by mapping a data pattern on an independent section of a number line. <P>SOLUTION: In the data classification device, a classification index function for mapping a teacher pattern on the independent section of the number line by belonging classification is trained. In some cases, classification boundary information is also trained together. The classification index function constitutes a classification estimation function together with the classification boundary information. A classification boundary is a boundary between the independent sections. The independent section is defined as a classification section. Minimization of a classification error and maximization of a margin are included in targets of the training. A function of the classification estimation function is high-speed specification of a belonging classification section of output of the classification index function from the classification boundary information when estimating the belonging classification of an input pattern. The classification represented by the predetermined classification section is estimated as the belonging classification of the input pattern. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a data classification method, a data classification device, and a program.

Support vector machine (SVM: Support)
Vector
(Machine) (see Non-Patent Document 1) is a binary classifier with high generalization ability, but when it is extended to multi-value classification, slow processing speed is a bottleneck. The basic idea of SVM is depicted in FIG. Input data includes two types of patterns. SVM creates a hyperplane between two patterns through training. The distance from this hyperplane to the input pattern is maximized. In FIG. 1, “x” and “o” represent two types of input patterns, a solid line represents a hyperplane, and a dotted line represents a margin hyperplane representing the shortest distance (called a margin) from the input pattern to the hyperplane. Represents a virtual hyperplane. The shortest distance adjusted by a slack variable representing an error is also called a margin.

As a multi-value classification algorithm (see Non-Patent Document 2, Patent Document 1, and Patent Document 2) using a space division method, representative ones are the onevson method, onevsrest method, pairwise method, DDAG (Decision)
Directed Acyclic
(Graph) method, MOF (Multiclass Objective Function) method, and the like.
There is a multi-valued classification regression method similar to SVM (see Non-Patent Document 3) other than spatial partitioning.

In the present invention, the classification estimation function refers to a calculation mechanism that estimates or determines the affiliation classification of an input pattern.
In the present invention, the classification index function refers to a calculation mechanism that converts a value for estimating the affiliation classification of an input pattern from the input pattern.
All the above-mentioned various methods construct a plurality of classification index functions, construct a classification estimation function by a combination of classification index functions, and estimate the affiliation classification of the input pattern.

The MOF method creates the relationship between all input data as follows. For one teacher pattern, it is constrained that the output of the classification index function of the affiliation classification is larger than the output of the classification index function of all the non-affiliation classifications. Constrain n functions simultaneously for n classifications. Since there are n (n-1) relationships among n functions, the number of constraints is also n (n-1) times. When estimating the affiliation classification of the input pattern, the classification represented by the function with the largest output is selected as the affiliation classification. The MOF method is the most ideal method, but its huge problem scale is difficult to put into practical use.
The MOF method can be considered as an integrated method of the onesvone method.

  The methods other than the MOF method and multi-value regression (hereinafter referred to as “synthetic training methods”) can be considered as decomposition for the MOF method. (N-1) or more binary classification estimation functions are constructed during training. However, the constraint relationship between functions is not obtained during training. We hope the result approximated to MOF method through the combination method after training. By reducing the constraints during training, the total time required for training is shortened. Note that by converting a single large-scale problem into a single row of small-scale problems, the demands on the capabilities of the computing equipment can be reduced. In the decomposition method, independent binary classification training is performed several times for one input data set, and the total size of the problem is proportional to the total number of classifications. Since the relationship between the binary classification estimation functions is difficult to constrain during training, the optimality of subsequent combinations also has a problem that is difficult to guarantee during training.

  Multi-valued regression hopes that the sum of deviations from the classification target points is minimized by defining the classification target points in space. The problem size is smaller than the various methods described above.

Japanese Patent Laid-Open No. 2004-280712 JP 2008-217375 A

Bernhard Scholkopf and Alexander J. Smola Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. The MIT Press. 2002. Mahesh Pal Multiclass Approaches for Support Vector Machine Based Land Cover Classification.MapIndia 2005 Conference Pei-Chun Chen, Tsung-Ju Lee, Yuh-Jye Lee, Su-Yun Huang, Multiclass Support Vector Classification via Regression.http: //www.stat.sinica.edu.tw/syhuang/papersdownload/MC-reg-121006 .pdf

Although the accuracy of the MOF method is expected to be high, a training problem with a scale of Θ ((mn) ² ) is formed for an input data set belonging to n classification and having m patterns. In a real data classification task, m is always much larger than n because multiple patterns always belong to one classification. Although depending on the ability of the computing equipment, training by the MOF method becomes difficult when n is not particularly small.

In various synthetic training methods, at least a training problem with a scale of Θ (m ² n) is formed. Although the scale may be smaller than that of the MOF method, if n is large, the computation time becomes considerably long. The speed can be increased by parallel processing by a plurality of computing devices, but the cost of the computing devices also increases.

  The optimal meaning of various synthetic training methods is obscured by the difference in interpretation of subsequent combinations. Therefore, it is difficult to achieve the goal of maximizing the overall margin. Since the size of the classification estimation function is optimized separately during training, there is a problem that the overall size optimization cannot be suppressed.

  The MOF method, various synthetic training methods, and multivalued regression calculate the outputs of a plurality of classification estimation functions for one input pattern in order to estimate the affiliation classification. The number of classification estimation functions is Θ (n). When N is large, calculation using many classification estimation functions becomes inefficient.

  Multilevel regression also has some other problems. First, since the margin is not defined, it is difficult to include distance maximization between classifications directly in the training target. Secondly, a plurality of functions are created by training as in the above-described various methods, and it is necessary to obtain and compare a plurality of function values for one pattern. 3. Since the output distribution of each function obtained by training is unknown, there is a possibility that the method of obtaining the probability from a single distance calculation method is insufficient.

  In order to solve the above-described problems caused by the prior art, a data classification device and a data classification method for constructing a single multi-level classification index function with a smaller scale operation and quickly identifying an input pattern classification based on the output of the classification index function And a program for causing a computer to execute the method.

The present invention uses a map as shown in FIG. 2 in order to solve the above problems. The map is implicit
mapping) or explicit mapping
mapping).

A data classification method, a data classification device, and a program according to the present invention include a step of dividing a single number line into a plurality of independent sections,
Setting the teacher data to belong to the independent section according to the affiliation classification;
A step for constructing a classification index function for mapping the teacher data to the section to which the teacher belongs or an approximation thereof by an undetermined coefficient method;
A training step with the goal of maximizing the margin attached to the boundary of the section;
Obtaining the classification index function by solving a mathematical programming problem;
It is characterized by including.

  Create objective programming and constraints, and create mathematical programming problems.

  Solve the mathematical programming problem.

  The classification index function and the classification boundary are determined from the solution of the mathematical programming problem, and a classification estimation function is created.

  Classify unknown data using a classification estimation function.

According to the present invention, the training problem scale of the multi-level classification method including error minimization and margin maximization as training targets can be reduced to O (m ² + mn).

  According to the present invention, only one classification index function is created by only one training, and the amount of training and the space required for the result are reduced.

  According to the present invention, with respect to the classification estimation of one input pattern, the output of the classification index function is obtained only once, and the amount of calculation of classification estimation based on the output of the classification index function is O (log (n)). The amount of execution of classification estimation is reduced. At the same time, there is no need for compounding (combination) of classification index functions, so there is no ambiguity.

  According to the present invention, compared with the regression method, the margin between the classifications can be defined, and each can be automatically adapted.

FIG. 1 is an explanatory view showing a conventional SVM training result. FIG. 2 is an explanatory view showing the solving means of the present invention. FIG. 3 is an explanatory diagram showing a training method for the multilevel classification apparatus of the present invention. FIG. 4 is an explanatory diagram showing a method for performing classification estimation using the multi-level classification apparatus of the present invention. FIG. 5 is an explanatory view showing an example of a classification boundary using the multilevel classification apparatus of the present invention.

<Training Embodiment>
The training embodiment is an embodiment for creating a classification estimation function for performing multi-value classification according to the present invention. FIG. 3 is an explanatory diagram showing a general process of the training embodiment. In the following, data will be described in pattern units.
G101, G102, G103, and G104 provide respective outputs to G105, a mathematical programming problem is formed in G105, solved in G108, and necessary information of the classification estimation function is prepared in G109 and G110. A classification estimation function is formed. The following describes each part in detail.

又は近似した機能を有する計算仕組みである。数１に、ｗをある空間直線の方向ベクトルとして、Φを入力パターンｘを高次元空間に写像する関数として、ｂをオフセットとして考えられる。数１にｗとｂは訓練の結果として得られるものである。Φは訓練の前に決定される関数である。ｆの出力は実数なので、結果としてｘは数直線に写像されることになる。 G101 is a mapping selection unit. The goal of map selection is to determine the classification index function in the form of undetermined coefficients. An example of a classification index function is

Or, it is a calculation mechanism having an approximate function. In Equation 1, w can be considered as a direction vector of a certain space straight line, Φ as a function for mapping the input pattern x to a high-dimensional space, and b as an offset. In Equation 1, w and b are obtained as a result of training. Φ is a function determined before training. Since the output of f is a real number, x is mapped to a number line as a result.

のように展開されることができる。 When Φ is defined as an explicit function, Φ receives x and outputs an N (N ≧ 1) -dimensional vector. If Φ _t (x) is the t-th term (t = 1... N) of Φ (x), f is

Can be deployed as follows.

である。カーナル関数を用いて、ｆは

のように展開されることができる。ｓ＝１…ｍ
。α_ｓは実数である。ｘ_ｓは第ｓ個教師パターンである。数４に、係数の１／２をα_ｓに合併しても、ｆは同等能力を有するため、ｆは

のように展開されても良い。 If Φ is defined as an implicit function, it can be defined by a kernel function. Some examples of kernel functions are

It is. Using the Kernall function, f is

Can be deployed as follows. s = 1 ... m
. α _s is a real number. x _s is the s-th teacher pattern. Even if ½ of the coefficient is merged with α _s in Equation 4, f has the same ability, so f is

It may be developed as follows.

  G102 expresses all teacher patterns in the form of vectors.

G103 is a classification boundary information designation unit.
The purpose of specifying classification boundary information is to select a means for acquiring section boundaries to constrain the output obtained by converting the teacher pattern belonging to each classification with a classification index function to sections for each classification.
There are a plurality of methods for specifying the classification boundary information. It is a prerequisite that there be a real set that is isomorphic to the set of intervals formed by all classification boundaries.
The following are some examples.

  When the classification boundary information is obtained from the mathematical programming solution, the classification boundary is made a variable, included in the mathematical programming problem composed of G105, and fixed as part of the output of G108.

When the classification boundary information is obtained from the solution of the mathematical program and other information, the classification boundary is made a variable while restricting the classification boundary information, and included in the mathematical programming problem composed of G105, and a part of the solution of the mathematical program. The information obtained by adjusting is used as the determined classification boundary information.
Unconstrained is one of the limiting means.
No adjustment is one of the adjustment means.

When the classification boundary information is designated as a constant, the classification boundary information that has become a constant is included in the mathematical programming problem configured by G105. The classification boundary information is obtained by adjusting the constant after the mathematical programming problem is solved.
No adjustment is one of the adjustment means.

  When the classification boundary information is calculated by a function that can be calculated backward, the classification identification symbol or equivalent is converted into a classification boundary value by a function that can be calculated backward.

  There is also a method of obtaining classification boundary information by combining the method for specifying each classification boundary information.

  G104 is a parameter designation unit. It is the purpose of G104 to adjust the relationship between the parts of the mathematical programming problem composed of G105. One example of parameter specification is to specify the weight of each part of the target function.

G105 constitutes a mathematical programming problem with the outputs of G101, G102, G103, and G104.
G105 is composed of two parts, G106 and G107.

G106 constitutes an objective function of the mathematical programming problem. The target function is a margin (a distance from a classification boundary to a map of the closest input pattern belonging to the classification in one classification section is called a margin. When a slack variable is used, the distance is adjusted by the slack variable. Approximation of maximization or margin maximization) is a partial target. The goal function has a partial goal of error minimization or error minimization approximation. The target function includes other partial targets as necessary.
An example of the “other partial target” is to minimize the difference between the minimum classification boundary and the maximum classification boundary.

G107 creates constraints for the mathematical programming problem. The constraint condition is that the output of the input pattern of the classification index function is not smaller than the lower margin point of the classification to which the pattern belongs (the value indicated by the margin from the classification boundary inside the classification section is called a margin point); It is constrained that it is not larger than the upper margin point of the classification of the pattern. When slack variables are used, the value after adjusting the output of the classification index function input pattern with slack variables is constrained. The constraint condition includes other constraint items as necessary.
An example of the “other restriction item” is to restrict the minimum classification section to a certain size or more.

  G108 solves the mathematical programming problem composed of G105.

  G109 separates information necessary for the classification index function from the output of G108, and determines the classification index function.

  G110 separates information for determining the classification boundary from the output of G108 as necessary, and determines the classification boundary.

G111 creates a classification estimation function using the output of G109 and G110. The classification estimation function may differ depending on the classification boundary designation method.
Regardless of the method for specifying the classification boundary, the classification estimation function searches the output of the classification index function for the input pattern into a set of classification sections constituted by the classification boundary, and is represented by the classification section including the output. There is a means of making the classification an estimated affiliation classification of the input pattern.
However, when the classification boundary is specified by a function that can be calculated backward, the classification of the input pattern can be estimated by back-calculating the output for the input pattern of the classification index function as the output of the function that can be calculated backward. Is possible.

<Embodiment of classification estimation>
The classification estimation embodiment is an embodiment for estimating the affiliation classification of an unknown input pattern using the classification estimation function created in G111. FIG. 4 is an explanatory diagram showing a general process of an embodiment of classification estimation.

  G201 is a classification estimation function providing unit. The classification estimation function providing unit provides the classification estimation function to the classification calculation unit.

  G202 is an unknown input pattern providing unit. The unknown input pattern providing unit provides an input pattern whose belonging classification is unclear as a vector to the classification calculation unit.

  G203 is a classification calculation unit. The classification calculation unit substitutes the unknown input pattern into the classification estimation function, and sets the output of the classification estimation function as the estimated affiliation classification of the input pattern.

  It should be understood that the examples described below are provided for illustrative purposes only and do not in any way define the scope of the invention.

となる。内、ｗ_ｔとｂは未定係数である。
Ｇ１０２は歩行のサンプルを特徴ベクトルに変換し、事前に得た歩行時患者の転倒可能性情報によって各サンプルに１から１０の分類番号を付ける。特徴ベクトルを、付けられた分類番号と共に、入力パターンとして提供する。前記ｘは当の特徴ベクトルのみである。
Ｇ１０３は−∞と０と１０と２０と３０と４０と５０と６０と７０と８０と＋∞を分類境界として提供する。以降の説明のため、前記の分類境界をｖ_０からｖ_１０で代表する。
Ｇ１０４はマージン最大化の部分目標に５０を重みとして、エラー最小化の部分目標に１００を重みとして提供する。
Ｇ１０６は

を目標関数として提供する。内、ξはエラーを記録するスラック変数のベクトルである。
Ｇ１０７は

を制約条件として提供する。
Ｇ１０５はＧ１０６とＧ１０７とを併せて

を訓練用の数理計画問題として提供する。内、ｖ_ｉは常数であり、ｘ_ｓ，ｔは第ｓ個教師パターンのベクトルの第ｔ項を意味し、ｘ_ｓはｖ_ｉ−１とｖ_ｉで確定された区間に代表された分類に所属し、１はｗの大きさに反比例するマージンである。
Ｇ１０８は専門ソフトで前記数理計画問題の解を提供する。
Ｇ１０９はＧ１０８で得た解からｗを獲得し、分類指標関数ｆを確定する。
Ｇ１１０はＧ１０３と同様の情報を提供する。
Ｇ１１１はＧ１０８で確定された分類指標関数ｆとＧ１１０で得た分類境界で作成された分類推定関数を提供する。当の分類推定関数は
ｆが負数を出力すれば分類番号１を出力し、
ｆが８０以上の値を出力すれば分類番号１０を出力し、
ｆが上記以外の値を出力した場合は該出力を１０で割った後の整数部分に１を足した後の値を分類番号として出力する。 Example 1 is an example of gait classification using secondary planning in the field of nursing care. The information provided to each unit in FIG. 3 is as follows.
G101 is the identity map of the positive map (identity
mapping). f is

It becomes. Of these, w _t and b are undetermined coefficients.
G102 converts the walking sample into a feature vector, and assigns a classification number of 1 to 10 to each sample according to the information on the possibility of falling of the walking patient obtained in advance. The feature vector is provided as an input pattern along with the assigned classification number. The x is only the feature vector.
G103 provides −∞, 0, 10, 20, 30, 40, 50, 60, 70, 80, and + ∞ as classification boundaries. For the following description, it is represented by v ₁₀ the classification boundary from v _0.
G104 provides 50 as a weight for the partial target for margin maximization and 100 as a weight for the partial target for error minimization.
G106

As a goal function. Ξ is a vector of slack variables for recording errors.
G107

As a constraint.
G105 is a combination of G106 and G107

As a mathematical programming problem for training. Among, v _i is the constant, x _{s, t} to means t-th term of the vector of the s pieces teacher pattern, x _s is v _i-1 and v _i is represented to the determined interval by classification 1 is a margin that is inversely proportional to the size of w.
G108 is a specialized software that provides a solution to the mathematical programming problem.
G109 obtains w from the solution obtained in G108, and determines the classification index function f.
G110 provides the same information as G103.
G111 provides a classification estimation function created by the classification index function f determined in G108 and the classification boundary obtained in G110. This classification estimation function outputs classification number 1 if f outputs a negative number,
If f outputs a value of 80 or more, the classification number 10 is output,
When f outputs a value other than the above, a value obtained by adding 1 to the integer part after dividing the output by 10 is output as a classification number.

  Temporarily, the classification estimation function is stored in a portable observation device. It is assumed that observations have been made.

The information provided to each part in FIG. 4 is as follows.
G201 reads the classification estimation function stored in the observation device and provides it to G203.
G202 provides an isomorphic vector having characteristics similar to the teacher pattern provided to G102 to G203 from the observed walking information of the patient.
G203 substitutes the vector provided to G202 into the classification estimation function provided to G201, and calculates the belonging classification of the walking information.

となる。内、α_ｓは未定係数であり、ｋは

である。
Ｇ１０２は予測される各の価格情報を分断し、事前調査によって確定された各時点の株価を記録して特徴ベクトルとする。分断の最後の日の平均株価と当の日の前日の平均株価との比例を１０段階に分けて１０分類とする。前記特徴ベクトル毎に一つの前記分類を所属分類として指定する。
前記特徴ベクトルと指定された前記分類とを一緒に教師パターンとして提供する。前記ｘは当の特徴ベクトルのみである。
Ｇ１０３は分類境界をｖ_０からｖ_１０の実数変数としての分類境界として提供する。但しｖ_０を０に指定する。
Ｇ１０４はマージン最大化の部分目標にＢを重みとして、エラー最小化の部分目標にＣを重みとして、分類境界制約の部分目標にＶを重みとして、Ｇ１０２に提供された全ての特徴ベクトルの各項の標準偏差の平均値をδとして提供する。同時に各教師パターンに起されるエラーの総エラーに対しての影響因子をベクトルｗとして提供する。
Ｇ１０６は

を目標関数として提供する。内、ξはエラーを記録するスラック変数のベクトルである。
Ｇ１０７は

を制約条件として提供する。
Ｇ１０５はＧ１０６とＧ１０７とを併せて

を訓練用の数理計画問題として提供する。内、ｖ_０は常数であり、ｖ_１からｖ_１０は変数であり、ｘ_ｓは第ｓ個教師パターンのベクトルを意味し、ｘ_ｓはｖ_ｉ−１とｖ_ｉで確定された区間に代表された分類に所属し、１はαの大きさに反比例するマージンである。
Ｇ１０８は専門ソフトで前記数理計画問題の解を提供する。
Ｇ１０９はＧ１０８で得た解からαを獲得し、分類指標関数ｆを確定する。
Ｇ１１０はＧ１０８で得た解からｖを獲得し、分類境界を確定する。
Ｇ１１１はＧ１０８で確定された分類指標関数ｆとＧ１１０で得た分類境界ｖで作成された分類推定関数を提供する。当の分類推定関数は、ｆの出力を含む、ｖ_ｉ−１とｖ_ｉで形成された実数区間を特定する。特定された区間に示された分類を出力する。 The second embodiment is an example of predicting the average stock price on the next day using linear programming in the financial field. The information provided to each unit in FIG. 3 is as follows.
G101 is a Gaussian
kernel). f is

It becomes. Where α _s is an undetermined coefficient and k is

It is.
G102 divides each price information to be predicted, and records the stock price at each time point determined by the preliminary survey as a feature vector. The proportion of the average stock price on the last day of the division and the average stock price on the previous day is divided into 10 levels and classified into 10 categories. For each feature vector, one classification is designated as the belonging classification.
The feature vector and the specified classification are provided together as a teacher pattern. The x is only the feature vector.
G103 is provided as a classification boundary as real variables of _{v 10} the classification boundary from _{v 0.} However, the _{v 0} to specify to zero.
G104 uses B as the weight for the margin maximization partial target, C as the weight for error minimization partial target, and V as the classification boundary constraint partial target for each term of all feature vectors provided to G102. The average value of the standard deviation is provided as δ. At the same time, an influencing factor for the total error of errors caused in each teacher pattern is provided as a vector w.
G106

As a goal function. Ξ is a vector of slack variables for recording errors.
G107

As a constraint.
G105 is a combination of G106 and G107

As a mathematical programming problem for training. V ₀ is a constant, v ₁ to v ₁₀ are variables, x _s means a vector of the s-th teacher pattern, and x _s is represented by an interval determined by v _i−1 and v _i. 1 is a margin that is inversely proportional to the size of α.
G108 is a specialized software that provides a solution to the mathematical programming problem.
G109 obtains α from the solution obtained in G108, and determines the classification index function f.
G110 obtains v from the solution obtained in G108, and determines the classification boundary.
G111 provides a classification estimation function created using the classification index function f determined in G108 and the classification boundary v obtained in G110. This classification estimation function specifies a real number interval formed by v _i−1 and v _i including the output of f. The classification shown in the specified section is output.

  Temporarily, the classification estimation function is stored in an analysis computer. It is assumed that analysis has been performed.

The information provided to each part in FIG. 4 is as follows.
G201 reads the classification estimation function stored in the computer and provides it to G203.
G202 provides an isomorphic vector having the same characteristics as the teacher pattern provided to G102 from the stock price data up to the day by the same division method as G102.
G203 substitutes the vector provided to G202 into the classification estimation function provided to G201, and calculates the affiliation classification of the stock price pattern indicated by the vector. The classification indicates the stock price forecast for the next day.

  It can be widely applied to all fields that use multi-value classification as a means. It is expected that many techniques based on multi-valued classification that could not be put into practical use due to the large scale of calculation until now can be put into practical use.

G101 Mapping selection unit G102 Teacher pattern providing unit G103 Classification boundary information designation unit G104 Parameter selection unit G105 Conversion unit to mathematical programming problem G106 Target function generation unit G107 Restriction condition generation unit G108 Mathematical programming problem solving unit G109 Classification index function G110 Classification boundary information recording part G111 Classification estimation function creation part G201 Classification estimation function provision part G202 Unknown input pattern provision part G203 Classification operation part O (f) Complexity large O symbol. If f is sufficiently large, it indicates a range of functions that does not increase faster than f.
Θ (f) Complexity Θ symbol. If f is sufficiently large, it indicates a function range in which the difference in increase speed from f is several times normal.

Claims (3)

A multi-value classification method for classifying input data into one of a plurality of classifications based on given teacher data,
Dividing a single number line into a plurality of independent sections;
Setting the teacher data to belong to the independent section according to the affiliation classification;
A step for constructing a classification index function for mapping the teacher data to the section to which the teacher belongs or an approximation thereof by an undetermined coefficient method;
A training step with the goal of maximizing the margin attached to the boundary of the section;
Obtaining the classification index function by solving a mathematical programming problem;
A data classification method characterized by including: A multi-value classification device for classifying input data into one of a plurality of classifications based on given teacher data,
Dividing a single number line into a plurality of independent sections;
Setting the teacher data to belong to the independent section according to the affiliation classification;
A step for constructing a classification index function for mapping the teacher data to the section to which the teacher belongs or an approximation thereof by an undetermined coefficient method;
A training step with the goal of maximizing the margin attached to the boundary of the section;
Obtaining the classification index function by solving a mathematical programming problem;
A data classification device comprising:   A program for causing a computer to execute the method according to claim 1.

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